Proof: Consider the partitioned matrix

$$\mathbf{M}=\left[\begin{array}{cc}\mathbf{A}& \mathbf{I}\\ \mathbf{I}& \mathbf{B}\end{array}\right].$$

Noting that $\mathbf{M}$ is positive definite if $\mathbf{A}\succ \mathbf{B}$, we can conclude that $({\mathbf{M}}^{-1}{)}_{22}$ is also positive definite (as is its inverse). Using the Schur complement, we therefore have

$${\mathbf{B}}^{-1}-\mathbf{I}{\mathbf{A}}^{-1}\mathbf{I}={\mathbf{B}}^{-1}-{\mathbf{A}}^{-1}\succ \mathbf{0}$$